# Full text of "Treatise On Analysis Vol-Ii"

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218 XIII INTEGRATION (a) Show that the following properties are equivalent: (a) The sequence (/xn) is convergent for the topology ^"e • (ft) For each closed subset A of X, the sequence (/u,n(A)) has a finite limit. (y) The sequence (fxn) converges for the topology ^"3 and satisfies the following condition : (C5) For each compact subset K of X and each e > 0, there exists an open neighborhood U of K such that | jLtn|(U — K) g e for all n. (8) The sequence (/xn) converges for the topology ^"3 and satisfies the following condition : (C6) For each e > 0 there exists 8 > 0 such that, for each universally measurable set A satisfying v(A) ^ 8, we have |^i,|(A) < B for all n. (To show that (/3) implies (y), use Problem 1 of Section 13.14 to show that the sequence ||//,n|| is bounded. To establish (C5), argue by contradiction: first consider the case in which the sequence (pn) tends to 0 for the topology ^"3 , and then pass to the general case as in Problem l(d). Next show that (y) implies that the sequence (/zn) is convergent for the topology &~s , and in particular implies (jS). To show that (8) implies (a), use the definition of measurable functions. To show that (a) implies (8), argue by contradiction. Finally, to show that (y) implies (a), consider first the case where lim ||jun|| = ||ju,||, where ju is the limit of (fx,n) for the topology ^"5 , and argue by n-K» contradiction to prove that (y) implies (8). To pass to the general case, reduce to the situation where /z = 0, and argue by contradiction: we may assume that there exists a function /e E6 such that the sequence (jU,n(/)) has a limit =£0. On the other hand, we can pick a subsequence (finfc) of the sequence (/xn) such that the sequence (p.nk\ and hence also (ftn~fc), converges vaguely. Hence arrive at a contradiction, by observing that these two sequences also satisfy (Cs).) (b) Give an example of a sequence (/xn) of measures belonging to M£(X, v) which converges for the topology ^"4, v but not for the topology &"s (cf. Section 13.18, Problem (c) Take X = [0, 1]. Show that the topologies &"$ and f6 on M£(X) are distinct (cf. Section 12.15, Problem 2(c), and Section 13.11, Problem 3). (d) Extend the results of Problems 1 and 2 to bounded complex measures. 3. Let X be a locally compact space, JJL a positive measure on X. Let (gn) be a sequence of ju-integrable functions such that (1) the sequence (gn) converges in measure (Section 13.12, Problem 2) to a function g\ (2)' the sequence of measures (gn • IJL) converges for the topology 3~$ (Problem 2). Show that under these conditions the function g is /x-integrable and that the sequence (gn) tends to g in «^i(X, ft). 4. Let X be a compact space, p, a positive measure on X, and (/„) an orthonormal sequence in ^c(^» fO consisting of functions which are uniformly bounded on X. (a) Show that, for each function #e ^c<X /*)» tne sequence of numbers (#]/„) = 1 0(x)fn(x) dfji(x) tends to 0. (Reduce to the case where g is bounded and hence belongs to :S?£(X, /*).) (b) Let x0eX. For the series £(#|/,)/n(#o) to converge for every function g e .5?£(X» /x), it is necessary and sufficient that (in the notation of Section 13.17, Problem 2) the sequence of bounded measures Kn(jc0 , •) • ft should converge for the topology «^"e (Problem 2) to a bounded measure hXQ - p,; hence in particular we have /;(*o) - jhxo(x)fn(x) dtfx) for all n.pology «^~4, v (use (e)). Give an example of a